# Equation of Motion for Polyakov Action with a mass term

I am having trouble solving the equation of motion for a polyakov action with a mass term (specifically, I am off by a factor of 2). If I am given a Polyakov action and a matter action: \begin{eqnarray} S_p &=& -\frac{T}{2}\int d^2\xi \sqrt{-\gamma}\gamma^{ab}\eta_{\mu\nu}\partial_ax^\mu\partial_bx^\nu \\ S_{mat} &=& T\int d^2\xi \sqrt{-\gamma}m^2 x^{\mu}x_{\mu} \end{eqnarray} I find that the equation of motion from varying $x^\mu\rightarrow x^\mu + \delta x^\mu$ is $\left(\Box + 2m^2\right)x^{\nu}=0$ but the solution that I am given is $\left(\Box + m^2\right)x^{\nu}=0$; where $\gamma_{ab}$ is the string metric, $\eta_{\mu\nu}$ is the metric in the (3+1 Minkowski) target space, and $\Box\equiv \frac{1}{\sqrt{-\gamma}}\partial_a\left( \sqrt{-\gamma}\gamma^{ab}\partial_b \right)$. (After looking at this for hours) Is there a typo in the actions or solution as given, or am I missing something fundamental about varying the action?

• Which reference? Which page? Commented May 20, 2017 at 19:51
• arxiv.org/abs/1405.3297 (not exactly the same action, but same idea)
– Bob
Commented May 20, 2017 at 19:54
• Yes, there is missing a factor half in the mass term. Commented May 20, 2017 at 20:23
• so you mean my solution is correct?
– Bob
Commented May 20, 2017 at 21:25

Your problem is in your matter action you forgot a 1/2 factor in the tension, the action should be $$S = - \frac{T}{2} \int \mathrm{d}^2 \xi \sqrt{- \gamma} \left( \gamma ^{a b} \partial _a x^{\mu} \partial _b x^{\nu} \eta _{\mu \nu} - m^2 x^{\mu} x^{\nu} \eta _{\mu \nu} \right)$$